1. MEASURES OF CENTRAL TENDENCY The measures of central tendency are quantities that describe the “center” of a data set. These are also called AVERAGES. In statistics, the three principal measures of central tendency or averages are MEAN, MEDIAN and MODE. A DATA ARRAY is a data set whose elements are ordered from the lowest value to the highest. Every quantitative data set can be arranged into a data array. Example: If the data set is {45, 12, 25, 16, 32, 7, 16, 20, 29} The corresp. data array is: {7, 12, 16, 16, 20, 25, 29, 32, 45}

2. THE MODE A. For the data set: {7, 12, 16, 16, 20, 25, 29, 32, 45} The MODE of a data set is the value that occurs most often. If a data set has only one mode, it is said to be UNIMODAL; if it has two, BIMODAL; if it has more than two, MULTIMODAL. Example: mode = 16 B. For the data set: {12, 5, 2, 5, 19, 2, 17, 5, 10, 2, 3, 12, 8} mode = 2, 5 C. For the data set: {21, 15, 32, 8, 14, 22, 17, 19, 10, 24, 29} mode = none If a data set has no mode, we just write “none” or “n.a.” (and never ‘mode = 0’, because ‘0’ is a data value.)

3. THE MEDIAN The MEDIAN of a data set is the central or middle value of the corresponding data array. Example: A. For the data set: {12, 5, 2, 5, 19, 2, 17, 5, 10, 2, 3, 12, 8} The corresponding data array is: { 2, 2, 2, 3, 5, 5, 5, 8, 10, 12, 12, 17, 19 } (13 values) (7 th position) median = 5 B. For the data set: {22, 6, 13, 17, 13, 18, 20, 18, 12, 10} The corresponding data array is: { 6, 10, 12, 13, 13, 17, 18, 18, 20, 22 } (10 values) (5½ th position) median = (13+17)/2 = 15 (midpoint of 5 th and 6 th values)

4. If a data array has n elements, then the median is at the position. If the calculated position is not a whole no., the median is the midpoint of the two middle values. C. For the data array: {22, 22, 25, 29, 32, 37, 37, 41} The median is at position (8+1)/2 = 4.5 median = (29+32)/2 = 30.5 (the median is in between the 4 th and 5 th data values) D. For the data array: {8, 12, 12, 13, 13, 19, 23, 27, 28, 28, 31} The median is at position (11+1)/2 = 6 median = 19 In computing the median of a data set, the corresponding data array must be set up first.

5. THE MEAN The MEAN (or EXPECTED VALUE) is the sum of all data values divided by the total no. of data values. In symbols: mean = where X 1, X 2, …, X N are the data values and N is the total no. of values. The symbol for sample mean is ; for population mean, μ. (mu) The MEAN (or EXPECTED VALUE) is the sum of all data values divided by the total no. of data values. In symbols: mean = where X 1, X 2, …, X N are the data values and N is the total no. of values. The symbol for sample mean is ; for population mean, μ. (mu) The symbol (summation) means “get the sum of all” what- ever is beside it. The letter, X, usually denotes data values.

6. Example: A. For the sample data set: {12, 5, 2, 5, 19, 2, 17, 5, 10, 2, 3, 12, 8}1 The mean is: B. For the population data set: 47423335304149372932 41402223314556273244 294238392622474330 The mean is:

7. AVERAGES FOR GROUPED DATA A data set can be presented as a frequency distribution (also called grouped data). How do we compute averages for this? BoundariesFrequency 0.95 – 8.7512 8.75 – 16.5515 16.55 – 24.3513 24.35 – 32.158 32.15 – 39.955 39.95 – 47.857 Example: One-way commuting distance of 60 FEU students. In the computation of averages for grouped data, we use the class boundaries instead of the class limits.

8. The MODAL CLASS of a grouped data is the interval class with the highest frequency. There can be more than one modal class. BoundariesFrequency 0.95 – 8.7512 8.75 – 16.5515 16.55 – 24.3513 24.35 – 32.158 32.15 – 39.955 39.95 – 47.857 Example: One-way commuting distance of 60 FEU students. modal class = 8.75–16.55 Does the modal class contain the mode of the data set?

9. The MEDIAN CLASS of a grouped data is the interval class containing the median (in the position ). To get the median class, keep adding the frequencies starting from the first interval class until you exceed the median pos- ition. The last interval class added is the median class. Example: One-way commuting distance of 60 FEU students. BoundariesFrequency 0.95 – 8.7512 8.75 – 16.5515 16.55 – 24.3513 24.35 – 32.158 32.15 – 39.955 39.95 – 47.857 median position = (60+1)/2 = 30.5 Cumulative Freq. 12 + 15 = 27 27 + 13 = 40 median class = 16.55 – 24.35

10. The MEAN of a grouped data is calculated using the formula for a data set, assuming that each interval class contains only its midpoint for data, with the corresponding frequency. BoundariesFreq.Midpt.Assumed data values 0.95 – 8.7512 8.75 – 16.5515 16.55 – 24.3513 24.35 – 32.158 32.15 – 39.955 39.95 – 47.857 4.85 12.65 20.45 28.25 36.05 43.9 4.85, 4.85, 4.85,… (12x) 12.65, 12.65, 12.65,… (15x) 20.45, 20.45, 20.45,… (13x) 28.25, 28.25, 28.25,… (8x) 36.05, 36.05, 36.05,… (5x) 43.9, 43.9, 43.9,… (7x) The formula for the MEAN of a grouped data is: mean = where X m is the midpoint of a class interval f is the corresponding class frequency and N is the total no. of data values. The formula for the MEAN of a grouped data is: mean = where X m is the midpoint of a class interval f is the corresponding class frequency and N is the total no. of data values.

11. BoundariesFreq, fMidpt, X m f · X m 0.95 – 8.75124.85 8.75 – 16.551512.65 16.55 – 24.351320.45 24.35 – 32.15828.25 32.15 – 39.95536.05 39.95 – 47.85743.9 58.2 Example: One-way commuting distance of 60 FEU students. 189.75 265.85 226 180.25 307.3 Σf · X m = 1227.35 mean == 20.46 (kms)

12. LOCATION OF THE AVERAGES IN A HISTOGRAM 15161720 23242527 29 3031 3234 35363839 4041 42 4344 4548 4950 5658 Data set (array):Histogram: mode =34 median =34 mean =34.75 Usually, for bell-shaped distributions, the mean, median and mode are equal or almost equal.

13. 1415202123 252728 30 3234 35 36373841 4445 46474950 51525356 57 58 59 Data set (array):Histogram: mode =41 median =41 mean =40.075 Usually, for left-skewed distributions, the mean is less than the median and the mode.

14. 1415 16 1718202123 24 25 2728 30 31 3234 35 3637 4142 4445 4951535860 Data set (array):Histogram: mode =23 median =30 mean =31.3 Usually, for right-skewed distributions, the mean is greater than the median and the mode.